Base model update - independence in prevalence over space and time; species functional groups; sampling sites within watersheds.
\[y_{i,j,k,l} \sim \text{Bin}(p_{i,j,k,l}, n_{i,j,k,l}) \\
p_{i,j,k,l} = Se_l * \lambda_{i,j,k,l} + (1 - Sp_l) * (1 - \lambda_{i,j,k,l}) \\
\lambda_{i,j,k,l} \sim \text{Beta}(\alpha_{\lambda_{m,i,j,l}}, \beta_{\lambda_{m,i,j,l}}) \\
Se_l \sim \text{Beta}(\alpha_{Se}, \beta_{Se}) \\
Sp_l \sim \text{Beta}(\alpha_{Sp}, \beta_{Sp}) \\
\]
- Where \(y_{i,j,k,l}, n_{i,j,k,l}\) are the number of test postive birds and total number of birds in a sampling event.
- Subscripts represent the month i, year j, sampling event k, and species group l.
- Species groups have their own sensitivity and specificity, but becuase their values are unknown, the priors are consistent for all species.
The true, unobserved prevlaence for a sampling event k within watershed m is, \(\lambda_{i,j,k,l}\).
We assume that the true prevlance for a sampling event can be represented by a Beta( \(\alpha_{\lambda_{m}}, \beta_{\lambda_{m}}\)) distribution. The mean true prevalence for watershed m is, \(\pi_m = \frac{\alpha_{\lambda_{m}}}{\alpha_{\lambda_m} + \beta_{\lambda_m}}\). Here, we drop the temporal/species subscripts for simplicity.
Sampling events are nested within months and watersheds.
Base model - independence in prevalence over space or time; mallards only; watershed-level data
\[y_{i,j,k} \sim \text{Bin}(p_{i,j,k}, n_{i,j,k}) \\
p_{i,j,k} = Se * \pi_{i,j,k} + (1 - Sp) * (1 - \pi_{i,j,k}) \\
\pi_{i,j,k} \sim \text{Beta}(\alpha_{\pi}, \beta_{\pi})\\
Se \sim \text{Beta}(\alpha_{Se}, \beta_{Se})\\
Sp \sim \text{Beta}(\alpha_{Sp}, \beta_{Sp}) \]
- Where \(y_{i,j,k}, n_{i,j,k}\) are the number of test postive birds and total number of birds sampled in a watershed.
- Subscripts represent the month, i, year, j, and watershed, k.
- We relate the true, unobserved prevalence (\(\pi_{i,j,k}\)) to the apparent prevelce (\(p_{i,j,k}\)) based on test sensitivity and specificity, which we assume is constant over space and time.
iCAR model - independence in prevalence over time, autocorrelation in space; mallards only; watershed-level data
\[y_{i,j,k} \sim \text{Bin}(p_{i,j,k}, n_{i,j,k}) \\
p_{i,j,k} = Se * \pi_{i,j,k} + (1 - Sp) * (1 - \pi_{i,j,k}) \\
\text{logit}(\pi_{i,j,k}) = \mu_{\pi,i,j} + \alpha_k \\
\alpha_k \sim \text{MVN}(0, \tau * I * (I - W))\\
\mu_{\pi,i,j} \sim \text{Cauchy}(0, 1/2.5)\\
\tau \sim \text{Gamma}(0.1, 0.1)\\
Se \sim \text{Beta}(\alpha_{Se}, \beta_{Se})\\
Sp \sim \text{Beta}(\alpha_{Sp}, \beta_{Sp}) \]
- Where \(y_{i,j,k}, n_{i,j,k}\) are the number of test postive birds and total number of birds sampled in a watershed. Additional data is incorporated in the Weight Matrix, W, which defines if there is a spatial association or how much of an association there is.
- Spatial autocorrelation in (\(\pi_{i,j,k}\)) among watersheds is incorporated with the \(\alpha_k\) parameter.
- Note that the multivariate normal distribution is parameterized in jags as, \(\Omega\), the inverse of the variance-covariance matrix. with \(\tau = 1 / \sigma^2\).